Question

Compare the given number with the number \(e\). Is the number less than or greater than \(e\) ? $$ \left(1+\frac{1}{1,000,000}\right)^{1,000,000} $$

Short answer

The number \( \left(1+\frac{1}{1,000,000}\right)^{1,000,000} \) is approximately equal to Euler's number \( e \).

Step-by-step solution

Understand the Euler's number \( e \)

The number \( e \) is an essential constant in mathematics, approximately equal to 2.71828. It is the base of the natural logarithm and can be defined in several ways, one of them being the limit of \( (1 + 1/n)^n \) as \( n \) approaches infinity.

Evaluate the given expression

Although we do not have the capabilities to do the precise computation as \( n \) approaches infinity, we can evaluate the expression for the given value of \( n = 1,000,000 \). Since this value is large, the result will be close to \( e \). Using a calculator, the value for \( (1 + 1/1,000,000)^{1,000,000} \) is approximately 2.71828.

Compare the Result with \( e \)

Observing the estimated result from the previous step and the known value of Euler's number, we see that they are approximately equal.